Numerical computations of split Bregman method for fourth order total variation flow

Giga, Yoshikazu; Ueda, Yuki

doi:10.1016/j.jcp.2019.109114

Cited by 10 publications

(4 citation statements)

References 40 publications

(61 reference statements)

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“…As we have explained in the above, the computational cost of our method based on alternating split Bregman iteration is cheap. Indeed, as pointed out in [23] that if we choose parameters properly, the number of iterations in Algorithm (VP loc,split ; u (n−1) τ ) becomes small; however, as far as we know, there are no mathematical guideline on optimal choices of parameters (see [23,22] for detailed explanation on the choices of the initial values Z (0) and B (0) and the parameter ρ).…”

Section: Numerical Resultsmentioning

confidence: 99%

A new numerical scheme for constrained total variation flows and its convergence

Giga,

Sakakibara,

Taguchi

et al. 2019

Preprint

Self Cite

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In this paper, we propose a new numerical scheme for a spatially discrete model of total variation flows whose values are constrained to a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex; hence it is hard to calculate a minimizer of the functional with the manifold constraint even if it exists. We overcome this difficulty by "localization technique" using the exponential map and prove a finite-time error estimate. Finally, we show a few numerical results for the target manifolds are S 2 and SO(3). Total variation flow and Manifold constraint and Spatially discrete model

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Section: Numerical Resultsmentioning

confidence: 99%

A new numerical scheme for constrained total variation flows and its convergence

Giga,

Sakakibara,

Taguchi

et al. 2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…It has been proven to be equivalent to the augmented Lagrangian approach [82] and used in a variety of problems including image sensing [34,82], free boundary problems [28] and microstructure formation [36].…”

Section: Connection To Other Methodsmentioning

confidence: 99%

Accelerated computational micromechanics

Zhou,

Bhattacharya

2020

Preprint

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We present an approach to solving problems in micromechanics that is amenable to massively parallel calculations through the use of graphical processing units and other accelerators. The problems lead to nonlinear differential equations that are typically second order in space and first order in time. This combination of nonlinearity and nonlocality makes such problems difficult to solve in parallel. However, this combination is a result of collapsing nonlocal, but linear and universal physical laws (kinematic compatibility, balance laws), and nonlinear but local constitutive relations. We propose an operator-splitting scheme inspired by this structure. The governing equations are formulated as (incremental) variational problems, the differential constraints like compatibility are introduced using an augmented Lagrangian, and the resulting incremental variational principle is solved by the alternating direction method of multipliers. The resulting algorithm has a natural connection to physical principles, and also enables massively parallel implementation on structured grids. We present this method and use it to study two examples. The first concerns the long wavelength instability of finite elasticity, and allows us to verify the approach against previous numerical simulations. We also use this example to study convergence and parallel performance. The second example concerns microstructure evolution in liquid crystal elastomers and provides new insights into some counter-intuitive properties of these materials. We use this example to validate the model and the approach against experimental observations.

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“…Remark 5 As we have explained in the above, the computational cost of our method based on alternating split Bregman iteration is cheap. Indeed, as pointed out in [23], if we choose parameters properly, the number of iterations in Algorithm (VP loc,split ; u (n−1) τ ) becomes small; however, as far as we know, there are no mathematical guidelines on optimal choice of parameters (see [22,23] for detailed explanation on the choice of the initial values Z (0) and B (0) and the parameter ρ).…”

Section: Remarkmentioning

confidence: 99%

A new numerical scheme for discrete constrained total variation flows and its convergence

Giga

Sakakibara

Taguchi³

et al. 2020

Numer. Math.

Self Cite

View full text Add to dashboard Cite

In this paper, we propose a new numerical scheme for a spatially discrete model of total variation flows whose values are constrained to a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex; hence it is hard to calculate a minimizer of the functional with the manifold constraint even if it exists. We overcome this difficulty by “localization technique” using the exponential map and prove a finite-time error estimate. Finally, we show a few numerical results for the target manifolds $$S^2$$ S 2 and SO(3).

show abstract

Numerical computations of split Bregman method for fourth order total variation flow

Cited by 10 publications

References 40 publications

A new numerical scheme for constrained total variation flows and its convergence

A new numerical scheme for constrained total variation flows and its convergence

Accelerated computational micromechanics

A new numerical scheme for discrete constrained total variation flows and its convergence

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